FYI, there's no need for the patch. The game looks for the CD drive starting with drive letter A. If drive letter A isn't found then it just gives up immediately. Use Disk Management to map either your real optical drive or mounted ISO to A (doesn't matter where the media is, there just needs to exist some partition/drive mounted on A:).
I've been able to play with 0 crashes on Windows 10. The avi cutscenes don't play, but I'm wondering if that's a codec issue, as I can't play the .avi's video portion with Windows 10's default video player. (Update: Run the game in compatibility mode, e.g. Windows 95, to get the cutscenes to play on Windows 10.) The cutscenes do play on the Windows XP VM though (after adding a floppy drive, as A seems to be reserved there).
Randomly figured this out as I was testing it on my physical Windows XP box, and then wondering for a few hours just how it could possibly be working there with no problems while having issues with my Windows XP and Windows 98 VM.
Update: Decided to check back here and saw OPs edit. Thanks Ninja-Trix for the compatibility mode find! Didn't think of messing around with that to get cutscenes working; now I no longer need the Windows XP VM anymore.
IMO I'm kinda surprised nobody else figured this out. I'm guessing a combination of there being a patch + a very small community means that the probability of someone trying to explore a non-patch way to run the game was very unlikely. I'm not sure of the best way to add this information to this wiki I found when trying to get it to run (https://www.pcgamingwiki.com/wiki/Lego_Chess), but I did put the info in the discussion section: https://www.pcgamingwiki.com/wiki/Talk:Lego_Chess